Radishes are one thing I have some real experience in!
Do you have any pictures of the damage caused by the worms? Or even of the worm itself? There are a few different kinds.
Also, I'm growing white radish and cherry radish (right next to each other) and even that slight difference brings different bugs. The white radish leaves are far more prone to bugs and worms (and leaf miners! UGH) than the darker, tougher, pricklier cherry radish leaves. I also have spinach growing in that same location. The aphids love the spinach but the worms and leaf miners tend to leave my spinach alone.
You could get some neem oil and spray the soil pretty well before replanting. There's also a post somewhere around this site with advice on using hydrogen peroxide. I've also been spraying nematodes in the garden and establishing a strong population of green lacewings and ladybugs. On top of all that, if there is an area I'm concerned about, I will check it a few times each day and pick off the bugs and toss them into soapy water.
Good luck! I love using my radish greens.
